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arXiv:1307.4052 · Journal of Mathematical Physics

Aspects of the inverse problem for the Toda chain

Karol Kajetan Kozlowski

Published Dec 1, 2013• 11 citations
arXivHTMLPDF

Abstract

We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.

DOI
10.1063/1.4848778

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We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.
We reconstruct the canonical operators pi,qip_i,q_ipi​,qi​ of the quantum closed Toda chain in terms of Sklyanin's separated variables.
We consider quantum analogs of the relativistic Toda lattices and give new 2×22\times 22×2 LLL-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving one-dimensional separation equations.
We consider quantum analogs of the relativistic Toda lattices and give new 2×22\times 22×2 LLL-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving one-dimensional separation equations.